#8814041

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(2,5)(6,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26,31,28)(27,29,32,35)(30,33,39,36)(34,37,38,40);(3,7)(6,8)(10,14)(12,21)(15,17)(19,24)(25,27,31,32)(26,29,28,35)(30,38,39,34)(33,40,36,37);(1,2)(3,6)(4,5)(7,8)(10,14)(11,18)(16,20)(19,24)(25,28,31,26)(27,29,32,35)(30,36,39,33)(34,37,38,40);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,29)(26,32)(27,28)(30,37)(31,35)(33,38)(34,36)(39,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,29)(26,32)(27,28)(30,37)(31,35)(33,38)(34,36)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,30,31,39)(26,33,28,36)(27,34,32,38)(29,37,35,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8721237);